Stability and convergence of Nitsche H(div) formulations

Establish convergence and stability of the Nitsche-type unfitted bilinear form for the H(div) Riesz map and the associated Darcy problem with weakly imposed Dirichlet boundary conditions.

Background

The paper imposes Dirichlet boundary conditions weakly through Nitsche-type terms for the H(grad)-, H(curl)-, and H(div)-conforming formulations. It notes that convergence and stability are established for the H(grad) formulation and have recently been proved for the corresponding H(curl)-related formulation.

For the H(div) Riesz map and Darcy problem, however, the authors state that convergence and stability remain unresolved. The difficulty is described as also occurring for body-fitted meshes, so the issue is not solely a consequence of unfitted geometry.

References

To our knowledge, the convergence and stability of $\widehat{}$ remain open problems.

— Geometric multigrid preconditioners for unfitted discretisations of the de Rham complex  (2610.22865 - Badia et al., 16 Oct 2026) in Remark 2.14, Section 2, subsection “Essential boundary conditions”